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### Title Real Analysis (Code 09010501)

 S.N Unit Content of the topics Learning objects Teaching Guidelines Methodology Time (Hrs) 1 one Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. Improper integrals and their convergence, Comparison tests, Abel’s and Dirichlet’s tests, Frullani’s integral, Integral as a function of a parameter. Continuity, Differentiability and integrability of an integral of a function of a parameter. Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. Students will be able to 1. Divisibility, G.C.D., L.C.M.. 2. Primes and Fundamental theorem of Arithmetic, 3. Fermat’s Theorem, Wilson’s theorem and its converse. Lecture should be effective so that student can grasp the topics easily. Lecture.  Seminar. Discussion/Interaction with Students. Assignment.  Discussion on Assignment.         Evaluation of Assignment. 15 2 Two Definition and examples of metric spaces metrics Definition and examples of metric spaces metrics Definition and examples of metric spaces metrics Definition and examples of metric spaces Students will be able to 1. Complete residue system , Quadratic residue, Euler’s ø function 2. Generalization of Fermat’s Theorem.Chinese remainder Theorem Legendre symbols, Gauss integer function,Chinese remainder theorem. Lecture should be effective so that student can grasp the topics easily. Lecture.  Seminar. Discussion/Interaction with Students. Assignment. Discussion on Assignment.         Evaluation of Assignment 15 3 Three Continuous functions, uniform continuity, compactness for metric spaces, sequential compactness, Bolzano-Weierstrass property, total boundedness, finite intersection property, continuity in relation with compactness, connectedness , components, continuity in relation with connectedness. Continuous functions, uniform continuity, compactness for metric spaces, sequential compactness, Bolzano-Weierstrass property, total boundedness, finite intersection To make students familiar with Direct circular and hyperbolic functions and their properties, De Moivre’s Theorem and its applications. Inverse circular and hyperbolic functions and their properties, Logarithm of a complex quantity. Gregory series, Summation of Trigonometric series Lecture should be effective sothat student can grasp the topics easily. Lecture. Seminar. Discussion/Interaction with Students. Assignment. Discussion on Assignment.        Evaluation of Assignment 15